Finite Group Databasebeta

← #12360

Subgroups — #12360

Order 256.

154 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 41 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3389C211yesmaximal, characteristic, complemented
2#2244C211yesmaximal, characteristic, complemented
3#2206C211yesmaximal, characteristic
4#1688C211yesmaximal, characteristic, complemented
5#1363C211yesmaximal, characteristic, complemented
6#1338C211yesmaximal, characteristic, complemented
7#1333C211yesmaximal, characteristic, complemented
8#554—21no—
9C2×D16—21no—
10#501—21no—
11#499C2211yescharacteristic
12#493C2211yescharacteristic
13#478—21no—
14#478—21no—
15#474—21no—
16#460—21no—
17#446C2211yescharacteristic
18#442—21no—
19C4×D8C2211yescharacteristic
20C4⋊C16C2211yescharacteristic
21#357—21no—
22#357—21no—
23#357—21no—
24#346C2211yescharacteristic
25#334C2211yescharacteristic
26#331—21no—
27C2×SD16—41no—
28C2×D8—21no—
29C2×D8—21no—
30C2×D8D411yescharacteristic
31C4⋊Q8—21no—
32#127—21no—
33#125—21no—
34#125—21no—
35C22⋊Q8—41no—
36C22⋊Q8—41no—
37C22⋊Q8—41no—
38C4×Q8D411yescharacteristic
39C4×D4—21no—
40C4×D4—21no—
41C4×D4—21no—
42C2×C4⋊C4—41no—
43D16—42no—
44C2×C16—21no—
45C2×C16—21no—
46#107—21no—
47#107D411yescharacteristic
48#107D411yescharacteristic
49#107D411yescharacteristic
50#106—21no—
51#106—21no—
52C4⋊C8—21no—
53C4⋊C8—21no—
54C4⋊C8D411yescharacteristic
55#102—21no—
56#102—21no—
57#102—21no—
58#102—41no—
59#97—21no—
60C4×C8C2311yescharacteristic
61C2×Q8—41no—
62C2×Q8D811yescharacteristic
63C2×D4—21no—
64C2×D4—41no—
65C2×D4—41no—
66C2×D4—41no—
67C22×C4—41no—
68C22×C4—41no—
69C22×C4—41no—
70SD16—82no—
71D8—21no—
72D8—42no—
73D8—41no—
74D8—41no—
75D8SD1612yescomplemented
76C2×C8—21no—
77C2×C8—41no—
78C2×C8C2×D411yescharacteristic
79C2×C8C2×D411yescharacteristic
80C4⋊C4—21no—
81C4⋊C4—21no—
82C4⋊C4—21no—
83C4⋊C4—21no—
84C4⋊C4—41no—
85C4⋊C4—41no—
86C4⋊C4—41no—
87C4⋊C4—41no—
88C4⋊C4—41no—
89C4⋊C4D811yescharacteristic
90C22⋊C4—41no—
91C22⋊C4—41no—
92C22⋊C4—41no—
93C42—21no—
94C42C2×D411yescharacteristic
95C16—41no—
96C16—41no—
97C23—41no—
98C23—81no—
99Q8—41no—
100Q8—161no—
101D4—22no—
102D4—41no—
103D4—81no—
104D4—81no—
105D4—81no—
106D4—81no—
107D4—161no—
108C2×C4—21no—
109C2×C4—21no—
110C2×C4—41no—
111C2×C4—41no—
112C2×C4—41no—
113C2×C4—82no—
114C2×C4—161no—
115C2×C4—161no—
116C2×C4#12011yescharacteristic
117C2×C4C2×D811yescharacteristic
118C2×C4AGL(1,ℤ/8)11yescharacteristic
119C8—21no—
120C8—41no—
121C8—81no—
122C8C2×SD1611yescharacteristic
123C8#13711yescharacteristic
124C22—42no—
125C22—81no—
126C22—81no—
127C22—161no—
128C22—161no—
129C22—161no—
130C22#44711yescharacteristic
131C4—21no—
132C4—21no—
133C4—41no—
134C4—41no—
135C4—81no—
136C4—81no—
137C4—81no—
138C4#44811yescharacteristic
139C4#45411yescharacteristic
140C2—82no—
141C2—321no—
142C2#162711yescharacteristic
143C2#219011yescharacteristic
144C2#219211yescharacteristic
subgroupG/H
center, socleC22#447
commutator, Frattini, cosocleC4×C8C23
prime radical#2206C2
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ #446 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #3389C2semidirect
2. #2244C2semidirect
4. #1688C2semidirect
5. #1363C2semidirect
6. #1338C2semidirect
7. #1333C2semidirect
75. D8SD16semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#33892
#22442
#22062
#16882
#13632
#13382
#13332